A Designer's Guide to Lunar Far-Side Interferometer Array: Power Spectrum Measurement and Cosmological Constraints from the Dark Ages

The 21-cm emission line from neutral hydrogen during the cosmic Dark Ages can be a powerful probe of cosmological models and early universe physics. This work provides a quantitative forecast for the design requirements of a lunar far-side interferometer array aimed at measuring the 21-cm power spectrum and constraining inflationary models through the running of the spectral index $\alpha_s$. During the Dark Ages, larger collapsed objects have not yet formed, allowing linear perturbation theory to remain valid down to much smaller scales than is possible in current large-scale structure or CMB surveys. We first validate this linearity assumption by quantifying the contribution of minihalos to the 21-cm signal. We then establish a generalized and flexible analytical framework for the baseline density distribution of interferometers that may consist of an arbitrary number of stations or sub-arrays. Incorporating a realistic noise model, we determine the configurations necessary to reach the detection threshold and demonstrate that distributing the total collecting area into multiple stations can improve the signal-to-noise ratio of the power spectrum at a tunable small scale of interest by up to two orders of magnitude. We then show that a lunar array requires at least $\sim30,000$ probed Fourier modes to achieve a constraint on inflation of $\sigma(\alpha_s) = 0.034$, a result competitive with the Planck 2018 results and capable of distinguishing between different inflationary scenarios. We quantitatively explain how thermal noise severely erodes modes at high redshifts and small scales -- scales previously considered easily accessible to Dark Ages observations in the literature -- and discuss the prospects for Dark Ages observations as a new and independent probe despite this limitation.